A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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4answers
39 views

Uncountable increasing family of $\sigma$-algebras

Could someone give an example of what an uncountable increasing family of $\sigma$-algebras $\{\mathcal{F}_t\}_{t\geq 0}$, $(\mathcal{F}_s \subset \mathcal{F}_t$ for $s<t)$ might look like? For ...
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0answers
11 views

What is the probability of no events in a Markov-modulated Poisson process?

Suppose I have a two-state continuous-time Markov chain $M$ with rate matrix $Q$. $$ Q = \begin{bmatrix} -q_{01} & q_{01} \\ q_{10} & -q_{10} \end{bmatrix} $$ Now consider a Poisson process ...
3
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0answers
11 views

Brownian motion: Strong Markov versus translation invariance

In the proof of the reflection principle in Durrett's textbook (Theorem 8.4.1), there's a step which I'm a little shaky on. Basically, this proof invokes the strong Markov property so to set this up ...
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0answers
8 views

Gaussian Process with explicit basis functions

I am considering the Gaussian process with explicit basis functions as discussed in the book (section 2.7): http://www.gaussianprocess.org/gpml/chapters/RW2.pdf Has anyone tried to derive formulas ...
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1answer
13 views

Martingale and independent increment

I know that in $L^2$ martingale a have independent increments. In particular that $\mathbb{E}[(X_m-X_n)^2]=\mathbb{E}[X^2_m-X^2_n]$ if X is a martingale. Does this extend also for general $p\geq 1$ in ...
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0answers
17 views

Simple Stratonovich product for physical system

I was reading a physical textbook and they used the "Stratonovich product" referred to $v_1 \circ dW_1 = \frac{1}{2}[v_1 + (v_1+dv_1)]dW_1$. I think this product is from the Stochastic process, thus ...
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0answers
14 views

Distribution of hitting time for two border brownian motion

I'm trying to find the distribution of hitting times for two border brownian motion with respect to both the hitting time AND which border is hit. Is this well defined? This is assuming $W_0=0$ with ...
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0answers
14 views

Smith's Key Renewal Theorem for Renewal Function

Consider a renewal process $(N_t)_{t \geq 0}$ and its renewal function $M(t):=\mathbb{E}[N_t]$ with interarrival distribution function $F$. One can show that $M$ satisfies the $(F,F)$-renewal ...
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0answers
46 views

Analytic solution to stochastic differential equations

I need help to to find the analytic solution (if it exists) of the following system of SDE. Usually, I use Matlab as software but in this case I'm unable to use it in order to figure out the problem. ...
2
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0answers
20 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...
7
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1answer
124 views

When does $\sum_{i=1}^{\infty} X_i$ exist for random sequences $\{X_i\}_{i=1}^{\infty}$?

Suppose $\{X_1, X_2, X_3, \ldots\}$ is an infinite sequence of random variables such that $E[X_i]=0$ for all $i$, and $E[X_iX_j]=0$ whenever $i \neq j$. Further suppose the variances $\sigma_i^2 = ...
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1answer
16 views

Change from stochastic exponential to exponential of Lévy process - Applebaum

In the book "Lévy Processes and Stochastic Calculus (2 edition)" of prof. Applebaum, Theorem 5.1.6 introduce how to change stochastic exponential to exponential of a Lévy process. I am not sure about ...
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0answers
13 views

Formula for running-time complexity

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$ Now it ...
2
votes
1answer
35 views

Independent increments of a Poisson process

In the following $\{X_t\}$ is a Poisson process. Assume that I've proved that $P(X_s=i,X_t-X_s=k)=P(X_s=i)P(X_t-X_s=k)$ so that the two events are independent, does it follow that ...
2
votes
1answer
27 views

Intuition about Blumenthal's 0-1 law

I'm studying Brownian motion from Durrett. I'm trying to understand what Blumenthal's 0-1 law really says about what Durrett calls the germ field, $\mathcal{F}_0^+$. Let $\mathcal{F}_t^+ = \cap_{s ...
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0answers
26 views

Pure death processes

If $P_n (t)=\Pr (N (t)=n)$ and $N (0)=a$, how can I show that in a pure death process $$P_{(a-1)}(t)=a (e^{\mu t }-1)e^{-a \mu t}.$$ I showed that $P_a(t)=e^{-a \mu t}$. In fact I want to show ...
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1answer
26 views

Probability of time between two events in a poisson process

Suppose people arrive at a certain place according to a poisson process with rate 10 per day. 1) What is the expected time until the arrival of 100 person. 2) What is the probability that ...
3
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1answer
64 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
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0answers
5 views

2 dimensional stochastic partial differential equation [on hold]

I am trying to code 2 dimensional stochastic heat equation ($u_t=u_{xx}+u_{yy}+dW(t,x,y)$) in matlab. But i am in confusion with $dW(t,x,y)$. Please suggest me that, how to code the stochastic term.
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1answer
27 views

What are some applications of stochastic processes and advances probabilities in real world? [closed]

One obvious field is Finance what are some others applications ?
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1answer
17 views

Closed communicating class

Let $P_{ij}$ a transition matrix, a class $C$ is closed if given two different states $i$ and $j$ $$i\in C, i\rightarrow j\Rightarrow j\in C$$ If a Markov Chain is irreducible the transition matrix ...
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2answers
61 views

Probability in a fixed die

I have that transition matrix is ...
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1answer
37 views

Why is $f(X_t)-\int_0^t Af(X_s) \, ds$ a martingale for a Markov process $(X_t)_{t \geq 0}$?

I think if $A$ is the usual generator for the Markov process $(X_t)_t$ $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}$$ then we get that for any "nice" $f$ the process ...
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0answers
20 views

differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)} w.r.t$ t?

Here $\phi(u,t)=E\{e^{iut}\} $ is a characteristic function, $x_t$ is Gaussian. Differentiating $\phi(u,t)=e^{(iu^T\hat{x_t}-\frac{1}{2}u^TP_tu)}$ w.r.t t the result is $\phi_u=\phi[i\hat{x}_t-P_tu]$
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1answer
46 views

Understanding the Markov property of Brownian motion

I'm trying to understand the Markov property for Brownian motions in full generality. The textbook I'm following states it like this: Recall that we have a family of measures $P_x, x \in ...
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0answers
36 views

expected value of expected value

I want to quantify the error of phase noise in terms of its normalized mean squared error. I define the error measure as (x is the error free function, y the distorted): $$ \rm NMSE = \frac{\int ...
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0answers
72 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
3
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0answers
26 views

Ito's formula and Infinitesmal generator

Consider an Ito process $$ dX_t = \sigma_t dB_t $$ where $\sigma_t$ is a two-state continuous-time Markov chain with state space $\{ \sigma_1, \sigma_2 \}$ that switches state with Poisson ...
2
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1answer
22 views

Compute the expected value of a brownian motion

Suppose $X(t)$ is a brownian motion. Compute $E[X(1)X(5)X(7)]$. I know that the brownian motion has independent increments, so if we could write $X(1)X(5)X(7)$ as such, then we could use the ...
2
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2answers
76 views

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any ...
2
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0answers
22 views

Expected response time of Continuous time Markov chain

I'm studying CTMC (Continuous Time Markov Chains). I came across the following slide I don't understand how they got $M(t+h) = M(t) + \alpha h + M(t)\lambda h - M(t) \mu h +o(h)$ Could anyone ...
2
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0answers
12 views

Stationary distribution for Markov process with non-exponential waiting times

Stochastic processes often are described in terms of transition rates where the length of time waited before a transition occurs is an exponential random variable. For example: $0\rightarrow 1$ at ...
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1answer
22 views

Conditional probability and disjoint events

If $\cup_{n=1}^\infty B_n=\Omega$ and $P(\Omega)=1$ then $\sum_{n=1}^\infty P(B_n)=1$, now $$P(A)=\sum_{n=1}^\infty P(A|B_n)P(B_n)=p\sum_{i=1}^\infty P(B_n)=p$$ If $X$ and $Y$ are independents ...
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0answers
42 views

How to calculate hitting probabilities for Brownian motion.

Given a standard Brownian motion with no drift, the PDF is... $${{1} \over {t^{3/2} \cdot \sqrt{2\cdot \pi}}} \cdot e^{-1/{2t}}$$ (Derived from the CDF $\int_{-\infty}^{f(t)/\sqrt{t}} {1 \over {2 ...
3
votes
3answers
47 views

Find $p_{ij}^{(n)}$ for the transition matrix

Let $$P=\begin{bmatrix}\frac{1}{3}&0&\frac{2}{3}\\\frac{1}{3}&\frac{2}{3}&0\\\frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}$$ find ...
1
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1answer
28 views

What is a martingale array - its definition and importance?

What is a martingale array? What is the importance of defining such an array, instead of using a martingale itself? A common example of this definition is a martingale difference array.
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2answers
33 views

Markovian systems: Why must controls be independent of state?

I am currently working my way through Probabilistic Robotics by Thrun, Burgard, and Fox. On p. 91, I encountered the following statement: The Markovian assumption implies independence between ...
0
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0answers
8 views

Generating Correlated Samples: Cholesky Decomposition of Correlation Matrix or Covariance Matrix? [duplicate]

I have multiple correlated stochastic processes and I would like to generate correlated samples of them. From my understanding, if I have my samples $Z$ and a Cholesky decomposition of their ...
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votes
1answer
25 views

finding transition matrix and probability of a gambler's ruin [closed]

A gambler has \$2. At each play of a game,he loses \$1 with a probability $q$ but wins \$1 with probability $p$. He stops playing if he loses \$2 or wins \$4. i)What is the transition matrix $P$ of ...
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0answers
44 views

Expected time to failure

A machine needs two types of components in order to function. We have a stockpile of $n$ type-$1$ components and $m$ type-$2$ components. Type-$1$ components last for an exponential time with ...
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0answers
13 views

4th order correlations of a delta-correlated random process

Say I have a complex random variable A(z) that is $\delta$-correlated, i.e. I have: $ \begin{align}\langle A(z) \rangle &= 0 \\ \langle A(z) A^*(z') \rangle &= \delta(z-z') \end{align} $ ...
0
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2answers
20 views

When does an uncountable collection of random variables define a stochastic process?

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $( \mathcal{X}, \mathcal{B})$ be a measurable space. Let $\{X_t\}_{t\in [0,1]}$ be an uncountable collection of random variables such ...
0
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0answers
25 views

Brownian bridge sde

The SDE for the Brownian bridge is the following: $dX_t = \dfrac{b-X_t}{1-t}dt+dB_t$ with the solution $X_t = a(1-t)+bt+(1-t)\int_{0}^t \dfrac{dB_s}{1-s}$. The expectation and covariance are: ...
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0answers
30 views

Probability of the same order

Let's consider a set $A$ and another set $B$ where $B \subset T$ . Conside another set $C= T \backslash B$(exclude set B from T). Now, We are given a stochastic process $X(t)$ such that $P(X(t)_{t \in ...
0
votes
2answers
23 views

Prove that a succession of random variables is a martingale

I've been working on the following problem: Let $\{{Y_n:n\in \mathbb{N}}\}$ be independent identically distributed random variables with mean $\mu$ and variance $\sigma^2>0$. Define ...
2
votes
3answers
31 views

Independent Poisson process

Suppose that $\{N_1(t),t\geq0\}$ and $\{N_2(t),t\geq0\}$ are independent Poisson Process with rates $\lambda_1$ and $\lambda_2$. Show that $\{N_1(t)+N_2(t),t\geq0\}$ is a Poisson process with ...
1
vote
1answer
24 views

Expectation and Poisson process

Let {$N(t),t\geq0$} be a Poisson process with rate $\lambda$. Calculate $E[N(t).N(t+s)]$ I know that $N(t)\sim Poisson(\lambda t)$ and $N(t+s)\sim Poisson(\lambda(t+s))$ I can assume that ...
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0answers
20 views

$\mathbb{F}$-Conditional density of a Brownian hitting time [closed]

Suppose that $(W_t)_{t\geq 0}\>$ is a standard Brownian motion on $(\Omega, (\mathcal{F}_t)_{t\geq 0 \>}, P)$ with $(\mathcal{F}_t)_{t\geq 0}\>$ the filtration generated by $W$. For $a\neq ...
2
votes
2answers
32 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...
1
vote
2answers
16 views

Effective inter-arrival time converge to mean

I am fairly new to statistics and just recently encountered queueing theory. I have programmed a simulation for a $M/M/1$ queue in which I specify the inter-arrival times and service times. I input ...